Question

question 5 of 25
stephanie invested money in an account where interest is compounded every year. she made no withdrawals or deposits.
the function $a(t) = 424(1 + 0.06)^t$ represents the amount of money in the account after $t$ years. how much money did stephanie originally invest?

a. $449.44
b. $476.41
c. $400.00
d. $424.00

Explanation:

To determine the original investment (principal amount) in a compound - interest problem, we use the compound - interest formula:

The general formula for compound interest is $A(t)=P(1 + r)^{t}$, where:

• $A(t)$ is the amount of money in the account after $t$ years.
• $P$ is the principal amount (the initial amount of money invested).
• $r$ is the annual interest rate (expressed as a decimal).
• $t$ is the number of years.

In the given function $A(t)=424(1 + 0.06)^{t}$, we compare it with the general compound - interest formula $A(t)=P(1 + r)^{t}$.

When $t = 0$ (at the time of the original investment, when no time has passed yet), we substitute $t = 0$ into the function $A(t)$:

Using the property of exponents, for any non - zero number $a$, $a^{0}=1$. So, when $t = 0$, $(1 + 0.06)^{0}=1$.

Then $A(0)=424\times(1 + 0.06)^{0}=424\times1 = 424$. But we can also directly identify from the formula $A(t)=P(1 + r)^{t}$ that the value of $P$ (the principal or the original investment) is the coefficient of $(1 + r)^{t}$. In the function $A(t)=424(1 + 0.06)^{t}$, the coefficient of $(1 + 0.06)^{t}$ is 424. This means that Stephanie originally invested $\$424.00$.

So the answer is D.

Answer:

To determine the original investment (principal amount) in a compound - interest problem, we use the compound - interest formula:

The general formula for compound interest is $A(t)=P(1 + r)^{t}$, where:

• $A(t)$ is the amount of money in the account after $t$ years.
• $P$ is the principal amount (the initial amount of money invested).
• $r$ is the annual interest rate (expressed as a decimal).
• $t$ is the number of years.

In the given function $A(t)=424(1 + 0.06)^{t}$, we compare it with the general compound - interest formula $A(t)=P(1 + r)^{t}$.

When $t = 0$ (at the time of the original investment, when no time has passed yet), we substitute $t = 0$ into the function $A(t)$:

Using the property of exponents, for any non - zero number $a$, $a^{0}=1$. So, when $t = 0$, $(1 + 0.06)^{0}=1$.

Then $A(0)=424\times(1 + 0.06)^{0}=424\times1 = 424$. But we can also directly identify from the formula $A(t)=P(1 + r)^{t}$ that the value of $P$ (the principal or the original investment) is the coefficient of $(1 + r)^{t}$. In the function $A(t)=424(1 + 0.06)^{t}$, the coefficient of $(1 + 0.06)^{t}$ is 424. This means that Stephanie originally invested $\$424.00$.

So the answer is D.